Show that if f (x) and g (x) are odd functions, then g ( f (x)) is odd.
To show that g(f(x)) is odd, we need to demonstrate that g(f(-x)) = -g(f(x)) for all values of x.
Since f(x) is an odd function, we have f(-x) = -f(x) for all x. Similarly, since g(x) is an odd function, we have g(-x) = -g(x) for all x.
Now, we can compute g(f(-x)) as:
g(f(-x)) = g(-f(x)) (using the fact that f(-x) = -f(x))
= -g(f(x)) (using the fact that g(-x) = -g(x))
Therefore, g(f(-x)) = -g(f(x)) holds for all values of x, showing that g(f(x)) is an odd function.